People are getting older and fatter, and many of them have money to spend on turning back the clock. Doctors, under pressure from insurance companies to push down prices, are searching for ways to collect more cash.

Put those trends together, and you’ve got a big business.

Hologic Inc., of Marlborough, whose products include breast cancer screening equipment and tests for sexually transmitted diseases, said Tuesday that it will pay $1.65 billion for Westford-based Cynosure Inc., a maker of fat-melting and skin-clearing lasers.

The deal, expected to close in the spring, would combine two Massachusetts manufacturers that approach the women’s health-and-beauty industry from decidedly different ends of the spectrum.

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